statistics, probability, nature of evidence

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Monthly Archives: December 2016

You may have encountered, in a book, article, or possibly a classroom, the now well-known game show puzzle sometimes called the Monty Hall problem. It goes like this: you are a contestant on a game show, presented with three closed doors. Behind one of the doors is a new car; the other two, goats.

Here’s how the game proceeds:

First, you choose any door you’d like.

Second, the host opens a door you didn’t choose, which (purposefully) reveals a goat.

Third, you have the option to switch your initial guess.

Then the door you finally settled on is opened: and you get whatever it reveals! The puzzle is this: to get the car, is it to your benefit to switch after the host opens a door? That is, will your odds of winning the car increase if you change your initial guess?

The mathematical solution to this problem is complicated to understand completely without a college-level understanding of probability. In fact, when the puzzle was first widely publicized by Marilyn vos Savant, it fooled hundreds of respected mathematicians and physicists. To this day, a basic internet search for the problem will turn up countless forum posts asking for help in understanding, and as many diverse attempts to explain the answer, often to no avail.

But the answer should be obvious, right? After the host reveals one of the goats, you must have a 50-50 shot at the car: it shouldn’t matter whether or not you switch doors. That turns out not to be correct: you’re actually twice as likely to take home the car if you switch your guess!

The issue people usually have with this puzzle is that the answer is non-intuitive: even if you understand the math behind the right answer, when you put yourself in the story, it doesn’t feel like switching should make a difference. The goal of this short post is to conjure the right gut reaction for this game – to call into being the intuition needed to “feel-out” the answer.

Consider a modification of the game in which there are not three but one hundred doors. There is still only 1 car, but now there are 99 goats. The rules can be generalized, like this:

First, you choose any door you’d like.

Second, the host purposefully reveals all but one goat (and that goat is behind either your door or the final door).

Third, you have the option to switch your initial guess.

As in the three-door game, by step three you are down to two options: either your initial guess was correct, or the final door has the car. Now, ask yourself: does this choice still feel like a 50-50 shot? In any 100-door game, it is quite likely (99% chance) that your first guess will be wrong, and therefore that the car is among the other doors. And, since the host will always end up opening all but one non-car door, the best strategy is to switch your guess.

The same principle applies in the 3-door game. More likely than not, your first guess was wrong: you only had a 1/3 chance of being right. After your initial guess, the host shows you all but one goat*. So, under a day-to-day interpretation of probability, there’s a 2/3rds chance the car is behind the final door.

*Notice that the 3-door game is the smallest game in which the host can take this action. In the 2-door game, it really is a 50-50 shot at the end: but that’s because your first guess is a 50-50 shot, and the host can do nothing.